Reserving day baths in Matsuyama

Hi, all! I'm going to Matsuyama soon, and want to experience the baths, but I need a private bath option (I'm trans). Just looking online, it's hard to tell which hotels and onsen offer reserved day baths.

Does anyone have any insight? Are they common? If my partner and I just walk around town a little and ask at various places, am we likely to be able to find a place to relax for an hour or so?

(I know it's possible to book a hotel room that comes with one, but owch, those prices!)

Thanks for any answers.